The signal induced by electrons and holes on the p(+) strips close to the interaction point can provide a very precise time mark of the photon interaction in a Silicon Drift Detector (SDD). A detailed study of the shape of the induced signals and of the parameters affecting the achievable time resolution is presented together with the first experimental tests on a linear SDD. The measured time resolution of the trigger signal is 1.5 ns r.m.s.. Such a fast trigger makes possible the use of SDDs as high resolution X-ray imagers and opens to the development of novel Compton telescopes for gamma-ray imaging based on Controlled Drift Detectors. (C) 2003 Elsevier B.V. All rights reserved.